E. Sevos'yanov | Searcharxiv

arXiv2022

Beltrami equations and mappings with asymptotic homogeneity at infinity

First of all, we prove that the BMO condition by John-Nirenberg leads in the natural way to the asymptotic homogeneity at the origin of regular homeomorphic solutions of the degenerate Beltrami equations. Then on this basis we establish a series of criteria for the existence of regular homeomorphic solutions of the degenerate Beltrami equations in the whole complex plane with asymptotic homogeneity at infinity. These results can be applied to the fluid mechanics in strictly anisotropic and inhomogeneous media because the Beltrami equation is a complex form of the main equation of hydromechanics.

math.CV↗

arXiv2022

Toward the theory of Dirichlet problem for the degenerate Beltrami equations

In this article, first we give a general lemma on the existence of regular homeomorphic solutions $f$ with the hydrodynamic normalization $f(z)=z+o(1)$ as $z\to\infty$ to the degenerate Beltrami equations $\overline{\partial}f=μ\,\partial f$ in $\mathbb C$ whose complex coefficients $μ$ have compact supports. On this basis, we establish criteria for existence and representation of regular discrete open solutions for the Dirichlet problem with continuous data to degenerate Beltrami equations in arbitrary simply connected bounded domains $D$ in $\mathbb C$. Moreover, we obtain similar criteria for the existence of multi-valued solutions $f$ in the spirit of the theory of multi-valued analytic functions in arbitrary bounded domains $D$ in $\mathbb C$ with no boundary component degenerated to a single point. Note that the latter request is necessary and that the real parts $u$ of such solutions $f$ are the so-called $A-$harmonic functions, i.e., single-valued continuous weak solutions of elliptice quations ${\rm div}\, (A\cdot\nabla u)=0$ with matrix-valued coefficients $A$ associated with $μ$. Thus, the results can be applied to potential theory in anisotropic and inhomogeneous media.

math.CV↗